Abstract
The main purpose of this paper is, using some properties of the Chebyshev polynomials, to study the power sum problems for the sinx and cosx functions and to obtain some interesting computational formulas.
Highlights
As is well known, the Chebyshev polynomials of the first kind {Tn(x)} and the Chebyshev polynomials of the second kind {Un(x)} are defined by T (x) =, T (x) = x, U (x) =, U (x) = x and Tn+ (x) = xT√n+ (x) – Tn(x), Un+ (x√) = xUn+ (x) – Un(x) for all integers n ≥
In a private communication with professor Wenpeng Zhang, he suggested us to give some explicit formulas for the following trigonometric power sums: q
2 Several simple lemmas To complete the proofs of our theorems, we need some new properties of Chebyshev polynomials, which we summarize as the following lemmas
Summary
If we take x = cos θ , sin((n + )θ ) In a private communication with professor Wenpeng Zhang, he suggested us to give some explicit formulas for the following trigonometric power sums: q– We shall use the properties of the Chebyshev polynomials of the first kind to obtain some closed formulas for the above trigonometric power sums. These results are stated in the following theorems.
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